We prove a new interchange theorem of infimum and integral. Its distinguishing feature is, on the one hand, to establish a general framework to deal with interchange problems for nonconvex integrands and nondecomposable sets, and, on the other hand, to link the theorems of Rockafellar and Hiai-Umegaki with the one of Bouchitté-Valadier. We give an application to relaxation of nonconvex geometric integrals of Calculus of Variations.
Acerbi, Buttazzo and Percivale gave a variational definition of the nonlinear string energy under the constraint "detu>0" (see [E. Acerbi, G. Buttazzo, D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity 25 (1991) 137-148]). In the same spirit, we obtain the nonlinear membrane energy under the simpler constraint "detu0".
We study the existence of an integral representation for the functionalwhen µ is a positive Radon measure on R N , Ω ⊂ R N is a bounded open set, and f : M m×N → [0, +∞[ is a continuous function not necessarily convex with growth conditions of order p > 1.
Consider a plate occupying in a reference configuration a bounded open set ⊂ R 2 , and let W : M 3×2 → [0, +∞] be its stored-energy function. In this paper we are concerned with relaxation of variational problems of type:= (x, 0) on ∂ } with p > 1, ·, · is the scalar product in R 3 and f ∈ L q ( ; R 3 ), with 1/ p + 1/q = 1, is the external loading per unit surface. We take into account the fact that an infinite amount of energy is required to compress a finite surface of the plate into zero surface, i.e., W ξ 1 | ξ 2 → +∞ as ξ 1 ∧ ξ 2 → 0.
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